With the rapid development of online medical platforms, consumer health questions (CHQs) are inefficient in diagnosis due to redundant information and frequent non-professional terms. The medical question summary (MQS) task aims to transform CHQs into streamlined doctors' frequently asked questions (FAQs), but existing methods still face challenges such as poor identification of question focus and model hallucination. This paper explores the potential of large language models (LLMs) in the MQS task and finds that direct fine-tuning is prone to focus identification bias and generates unfaithful content. To this end, we propose an optimization framework based on core focus guidance. First, a prompt template is designed to drive the LLMs to extract the core focus from the CHQs that is faithful to the original text. Then, a fine-tuning dataset is constructed in combination with the original CHQ-FAQ pairs to improve the ability to identify the focus of the question. Finally, a multi-dimensional quality evaluation and selection mechanism is proposed to comprehensively improve the quality of the summary from multiple dimensions. We conduct comprehensive experiments on two widely-adopted MQS datasets using three established evaluation metrics. The proposed framework achieves state-of-the-art performance across all measures, demonstrating a significant boost in the model's ability to identify critical focus of questions and a notable mitigation of hallucinations. The source codes are freely available at https://github.com/DUT-LiuChao/FocusMed.